diff --git a/Lecture 10/ANOVA Example.ipynb b/Lecture 10/ANOVA Example.ipynb new file mode 100644 index 0000000..85f2c6c --- /dev/null +++ b/Lecture 10/ANOVA Example.ipynb @@ -0,0 +1,394 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "ANOVA Pizza Case:\n", + "1 factor 3 level\n", + "In a good pizza restaurant, the three pizzaiolos argue about who bakes the best pizza. They bake 5 pizzas each, and invite 15 random gourmets to sample the pizza. The grade the taste numerically, such that higher numbers mean better taste.\n", + "\n", + "First, lets input the data. We will use a dataframe." + ] + }, + { + "cell_type": "code", + "execution_count": 51, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "'data.frame':\t15 obs. of 2 variables:\n", + " $ Taste: num 51 45 33 45 67 23 43 23 43 45 ...\n", + " $ Names: Factor w/ 3 levels \"Alberto\",\"Francesco\",..: 1 1 1 1 1 2 2 2 2 2 ...\n" + ] + } + ], + "source": [ + "mydataframe <- data.frame(\"Taste\" = c(51,45,33,45,67,23,43,23,43,45,56,76,74,87,56), \"Names\" = c(rep(\"Alberto\",5),rep(\"Francesco\",5),rep(\"Paolo\",5)))\n", + "str(mydataframe)" + ] + }, + { + "cell_type": "code", + "execution_count": 52, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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+ "text/plain": [ + "plot without title" + ] + }, + "metadata": { + "image/png": { + "height": 420, + "width": 420 + }, + "text/plain": { + "height": 420, + "width": 420 + } + }, + "output_type": "display_data" + } + ], + "source": [ + "plot(mydataframe$Taste)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We have 3 groups with n_i=5 points each, so the degrees of freedom for SST:3-1=2 and for SSE:15-3=12. Indeed, we find 12+2=14=n-1. \n", + "Now we need to compute the statistical parameters of the groups." + ] + }, + { + "cell_type": "code", + "execution_count": 57, + "metadata": {}, + "outputs": [], + "source": [ + "# Compute all variances = standard deviation ^2\n", + "s1sq=sd(mydataframe$Taste[1:5])^2\n", + "s2sq=sd(mydataframe$Taste[6:10])^2\n", + "s3sq=sd(mydataframe$Taste[11:15])^2\n", + "\n", + "# Compute all group averages\n", + "a1=mean(mydataframe$Taste[1:5])\n", + "a2=mean(mydataframe$Taste[6:10])\n", + "a3=mean(mydataframe$Taste[11:15])\n", + "\n", + "# Compute global average\n", + "a=mean(mydataframe$Taste)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Next we can compute SS, SST, SSE:" + ] + }, + { + "cell_type": "code", + "execution_count": 60, + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "4883.73333333333" + ], + "text/latex": [ + "4883.73333333333" + ], + "text/markdown": [ + "4883.73333333333" + ], + "text/plain": [ + "[1] 4883.733" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/html": [ + "3022.93333333333" + ], + "text/latex": [ + "3022.93333333333" + ], + "text/markdown": [ + "3022.93333333333" + ], + "text/plain": [ + "[1] 3022.933" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/html": [ + "1860.8" + ], + "text/latex": [ + "1860.8" + ], + "text/markdown": [ + "1860.8" + ], + "text/plain": [ + "[1] 1860.8" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "SS=sum((data-a)^2)\n", + "SST= 5*(a1-a)^2 + 5*(a2-a)^2 + 5*(a3-a)^2\n", + "SSE= (5-1)*s1sq + (5-1)*s2sq + (5-1)*s3sq\n", + "SS\n", + "SST\n", + "SSE" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Always test that SST+SSE=SS!!" + ] + }, + { + "cell_type": "code", + "execution_count": 61, + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "4883.73333333333" + ], + "text/latex": [ + "4883.73333333333" + ], + "text/markdown": [ + "4883.73333333333" + ], + "text/plain": [ + "[1] 4883.733" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "SST+SSE" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Now we compute the mean squares by normalizing the sum of squares by their respective degrees of freedom" + ] + }, + { + "cell_type": "code", + "execution_count": 62, + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "1511.46666666667" + ], + "text/latex": [ + "1511.46666666667" + ], + "text/markdown": [ + "1511.46666666667" + ], + "text/plain": [ + "[1] 1511.467" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/html": [ + "155.066666666667" + ], + "text/latex": [ + "155.066666666667" + ], + "text/markdown": [ + "155.066666666667" + ], + "text/plain": [ + "[1] 155.0667" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "MST=SST/2\n", + "MSE=SSE/12\n", + "MST\n", + "MSE" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Now we compute the F-statistic:" + ] + }, + { + "cell_type": "code", + "execution_count": 63, + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "9.74720550300946" + ], + "text/latex": [ + "9.74720550300946" + ], + "text/markdown": [ + "9.74720550300946" + ], + "text/plain": [ + "[1] 9.747206" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "F=MST/MSE\n", + "F" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "This is a large F, but to make a statistical statement we need to compare it to the quantile of the F-distribution. Large values of F disprove H0, so we find the maximal F at which we still accept H0 at a level of significance 0.05 as:" + ] + }, + { + "cell_type": "code", + "execution_count": 64, + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "3.88529383465239" + ], + "text/latex": [ + "3.88529383465239" + ], + "text/markdown": [ + "3.88529383465239" + ], + "text/plain": [ + "[1] 3.885294" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "F_crit=qf(0.95,2,12) #(q)uantile of the (f)-distribution at alpha, with df1 and df2 degrees of freedom.\n", + "F_crit" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Indeed, our F is significantly above the critical value, so we can reject the null hypothesis that all pizza are equivalent. Clearly there are better and worse bakers in the restaurant." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Let us redo all with internal R commands:" + ] + }, + { + "cell_type": "code", + "execution_count": 65, + "metadata": {}, + "outputs": [], + "source": [ + "aov.out=aov(Taste~Names, data=mydataframe)" + ] + }, + { + "cell_type": "code", + "execution_count": 66, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + " Df Sum Sq Mean Sq F value Pr(>F) \n", + "Names 2 3023 1511.5 9.747 0.00306 **\n", + "Residuals 12 1861 155.1 \n", + "---\n", + "Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "summary(aov.out)" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "R", + "language": "R", + "name": "ir" + }, + "language_info": { + "codemirror_mode": "r", + "file_extension": ".r", + "mimetype": "text/x-r-source", + "name": "R", + "pygments_lexer": "r", + "version": "3.6.3" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/Lecture 10/Auto ANOVA.ipynb b/Lecture 10/Auto ANOVA.ipynb new file mode 100644 index 0000000..78b2d7a --- /dev/null +++ b/Lecture 10/Auto ANOVA.ipynb @@ -0,0 +1,166 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 28, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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KFeGDD7I9BwAAAABJIoxS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6EYzbRSJ0ZzZcWgQcMUowAAAACdhmI000o3bIhevVKn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\n0tq1ayOiW7duDY5WVFSsWbOmb9++uYLNnz9/4MCB2xds0aJFu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+ "text/plain": [ + "Plot with title “Histogram of Fvals”" + ] + }, + "metadata": { + "image/png": { + "height": 600, + "width": 900 + }, + "text/plain": { + "height": 600, + "width": 900 + } + }, + "output_type": "display_data" + } + ], + "source": [ + "options(repr.plot.width=15, repr.plot.height=10) # this command just formats the size of the figures. Adapt to view them nicely\n", + " # in your browser.\n", + "\n", + "\n", + "# Automatic repetition: Lets do 1000 experiments and do an ANOVA for each of them\n", + "Fvals=rep(0,1000) # create an empty vector for all the F-values\n", + "\n", + "for (i in seq(1,1000,by=1)) # run 1000 experiments and record the F-value for it. We have 5 groups with 10 datapoints each\n", + "{\n", + "n=10\n", + "T1 = rnorm(n,mean=18,sd=3)\n", + "T2 = rnorm(n,mean=18,sd=2)\n", + "T3 = rnorm(n,mean=18,sd=4)\n", + "T4 = rnorm(n,mean=18,sd=2.5)\n", + "T5 = rnorm(n,mean=18,sd=3)\n", + "\n", + "datastructure = data.frame(T1,T2,T3,T4,T5) # creates a dataframe from the 5 vectors (2D data)\n", + "mydata = stack(datastructure) # Takes the data, labels it with group index, and generates one long stacked vector (1D data)\n", + "\n", + "analysis=anova(lm(values~ind,data=mydata))\n", + "Fvals[i]=analysis$`F value`[1] #Compute and store F-values each cycle\n", + "}\n", + "\n", + "histdata=hist(Fvals,ylim=c(0,100),breaks=seq(0,15,by=0.1))\n", + "\n", + "# Compare to distribution function\n", + "x=seq(0,6,by=0.01)\n", + "y=100*df(x,df1 = 4, df2 = 45) # F distribution if all had same means. The red line is the theoretical distribution function.\n", + "lines(x,y,col=\"red\") \n", + "abline(v=qf(p = 0.95,df1 = 4, df2 = 45),col=\"blue\") #Critial value 5%" + ] + }, + { + "cell_type": "code", + "execution_count": 29, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "List of 6\n", + " $ breaks : num [1:151] 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ...\n", + " $ counts : int [1:150] 36 48 85 53 58 78 73 54 51 40 ...\n", + " $ density : num [1:150] 0.36 0.48 0.85 0.53 0.58 0.78 0.73 0.54 0.51 0.4 ...\n", + " $ mids : num [1:150] 0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 ...\n", + " $ xname : chr \"Fvals\"\n", + " $ equidist: logi TRUE\n", + " - attr(*, \"class\")= chr \"histogram\"\n" + ] + } + ], + "source": [ + "str(histdata)" + ] + }, + { + "cell_type": "code", + "execution_count": 30, + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "2.57873918431156" + ], + "text/latex": [ + "2.57873918431156" + ], + "text/markdown": [ + "2.57873918431156" + ], + "text/plain": [ + "[1] 2.578739" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "qf(p = 0.95,df1 = 4, df2 = 45)" + ] + }, + { + "cell_type": "code", + "execution_count": 31, + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "0.955" + ], + "text/latex": [ + "0.955" + ], + "text/markdown": [ + "0.955" + ], + "text/plain": [ + "[1] 0.955" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "sum(histdata$counts[1:27])/sum(histdata$counts)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "R", + "language": "R", + "name": "ir" + }, + "language_info": { + "codemirror_mode": "r", + "file_extension": ".r", + "mimetype": "text/x-r-source", + "name": "R", + "pygments_lexer": "r", + "version": "3.6.3" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/Lecture 10/F-distribution.ipynb b/Lecture 10/F-distribution.ipynb new file mode 100644 index 0000000..9807bcc --- /dev/null +++ b/Lecture 10/F-distribution.ipynb @@ -0,0 +1,67 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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t8PNbiEqKMC/CB9FmR+3X5x3i7vgp+L4BZOgzSdtVqw17t3QStGub8qP0gLyxKW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\nBTpVQdq9ciXnu5wgPdWCoLoLPVgFa6HoTFWQZvzjYsNfQ4eUuNJ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+ "text/plain": [ + "plot without title" + ] + }, + "metadata": { + "image/png": { + "height": 420, + "width": 420 + }, + "text/plain": { + "height": 420, + "width": 420 + } + }, + "output_type": "display_data" + } + ], + "source": [ + "## F-distribution\n", + "# Divide 2 chi^2 RV. This appears in ratios of variances\n", + "\n", + "\n", + "nu1 = 3 #numerator df\n", + "nu2 = 12 #denumerator df\n", + "\n", + "x=seq(0,5,by=0.01)\n", + "y= df(x,df1 = nu1, df2 = nu2)\n", + "\n", + "plot(x,y,type = \"l\",xlab = \"F\",ylab = \"probability density\")" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "R", + "language": "R", + "name": "ir" + }, + "language_info": { + "codemirror_mode": "r", + "file_extension": ".r", + "mimetype": "text/x-r-source", + "name": "R", + "pygments_lexer": "r", + "version": "3.6.2" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +}